LMFDB - Space of modular forms of level 1568, weight 2, and Character 1568.i

Defining parameters

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.i (of order $3$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$ 7 $
Character field:			$\Q(\zeta_{3})$
Newform subspaces:			$ 25 $
Sturm bound:			$448$
Trace bound:			$25$
Distinguishing $T_p$:			$3$, $5$, $11$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(1568, [\chi])$.

	Total	New	Old
Modular forms	512	80	432
Cusp forms	384	80	304
Eisenstein series	128	0	128

Trace form

Decomposition of $S_{2}^{\mathrm{new}}(1568, [\chi])$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	$q$-expansion
1568.2.i.a	$2$	$12.521$	$\Q(\sqrt{-3}) $	None	$0$	$-2$	$-2$	$0$	$q+(-2+2\zeta_{6})q^{3}-2\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots$
1568.2.i.b	$2$	$12.521$	$\Q(\sqrt{-3}) $	None	$0$	$-2$	$0$	$0$	$q+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots$
1568.2.i.c	$2$	$12.521$	$\Q(\sqrt{-3}) $	None	$0$	$-2$	$0$	$0$	$q+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots$
1568.2.i.d	$2$	$12.521$	$\Q(\sqrt{-3}) $	None	$0$	$-2$	$2$	$0$	$q+(-2+2\zeta_{6})q^{3}+2\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots$
1568.2.i.e	$2$	$12.521$	$\Q(\sqrt{-3}) $	$\Q(\sqrt{-1}) $	$0$	$0$	$-4$	$0$	$q-4\zeta_{6}q^{5}+3\zeta_{6}q^{9}-4q^{13}+(-8+\cdots)q^{17}+\cdots$
1568.2.i.f	$2$	$12.521$	$\Q(\sqrt{-3}) $	$\Q(\sqrt{-1}) $	$0$	$0$	$-2$	$0$	$q-2\zeta_{6}q^{5}+3\zeta_{6}q^{9}-6q^{13}+(2-2\zeta_{6})q^{17}+\cdots$
1568.2.i.g	$2$	$12.521$	$\Q(\sqrt{-3}) $	$\Q(\sqrt{-1}) $	$0$	$0$	$2$	$0$	$q+2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+6q^{13}+(-2+\cdots)q^{17}+\cdots$
1568.2.i.h	$2$	$12.521$	$\Q(\sqrt{-3}) $	$\Q(\sqrt{-1}) $	$0$	$0$	$4$	$0$	$q+4\zeta_{6}q^{5}+3\zeta_{6}q^{9}+4q^{13}+(8-8\zeta_{6})q^{17}+\cdots$
1568.2.i.i	$2$	$12.521$	$\Q(\sqrt{-3}) $	None	$0$	$2$	$-2$	$0$	$q+(2-2\zeta_{6})q^{3}-2\zeta_{6}q^{5}-\zeta_{6}q^{9}+(4+\cdots)q^{11}+\cdots$
1568.2.i.j	$2$	$12.521$	$\Q(\sqrt{-3}) $	None	$0$	$2$	$0$	$0$	$q+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots$
1568.2.i.k	$2$	$12.521$	$\Q(\sqrt{-3}) $	None	$0$	$2$	$0$	$0$	$q+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots$
1568.2.i.l	$2$	$12.521$	$\Q(\sqrt{-3}) $	None	$0$	$2$	$2$	$0$	$q+(2-2\zeta_{6})q^{3}+2\zeta_{6}q^{5}-\zeta_{6}q^{9}+(-4+\cdots)q^{11}+\cdots$
1568.2.i.m	$4$	$12.521$	$\Q(\sqrt{-3}, \sqrt{5})$	None	$0$	$-2$	$-2$	$0$	$q+(-1-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots$
1568.2.i.n	$4$	$12.521$	$\Q(\sqrt{-3}, \sqrt{5})$	None	$0$	$-2$	$2$	$0$	$q+(-1-\beta _{1}-\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$
1568.2.i.o	$4$	$12.521$	$\Q(\sqrt{2}, \sqrt{-3})$	None	$0$	$-2$	$2$	$0$	$q+(-1+\beta _{1}-\beta _{2})q^{3}+(2\beta _{1}-\beta _{2}+2\beta _{3})q^{5}+\cdots$
1568.2.i.p	$4$	$12.521$	$\Q(\sqrt{-3}, \sqrt{7})$	None	$0$	$0$	$-6$	$0$	$q+\beta _{1}q^{3}+3\beta _{2}q^{5}+4\beta _{2}q^{9}+\beta _{1}q^{11}+\cdots$
1568.2.i.q	$4$	$12.521$	$\Q(\sqrt{2}, \sqrt{-3})$	None	$0$	$0$	$0$	$0$	$q+\beta _{1}q^{3}-\beta _{2}q^{9}+(-2-2\beta _{2})q^{11}+\cdots$
1568.2.i.r	$4$	$12.521$	$\Q(\sqrt{2}, \sqrt{-3})$	None	$0$	$0$	$0$	$0$	$q+\beta _{1}q^{3}-\beta _{2}q^{9}+(2+2\beta _{2})q^{11}-2\beta _{3}q^{13}+\cdots$
1568.2.i.s	$4$	$12.521$	$\Q(\sqrt{2}, \sqrt{-3})$	$\Q(\sqrt{-1}) $	$0$	$0$	$0$	$0$	$q+3\beta _{1}q^{5}+(3+3\beta _{2})q^{9}+5\beta _{3}q^{13}+\cdots$
1568.2.i.t	$4$	$12.521$	$\Q(\sqrt{2}, \sqrt{-3})$	$\Q(\sqrt{-1}) $	$0$	$0$	$0$	$0$	$q-\beta _{1}q^{5}+(3+3\beta _{2})q^{9}+\beta _{3}q^{13}+(-5\beta _{1}+\cdots)q^{17}+\cdots$
1568.2.i.u	$4$	$12.521$	$\Q(\zeta_{12})$	None	$0$	$0$	$2$	$0$	$q-\zeta_{12}^{2}q^{3}+(1-\zeta_{12})q^{5}+3\zeta_{12}^{2}q^{11}+\cdots$
1568.2.i.v	$4$	$12.521$	$\Q(\sqrt{-3}, \sqrt{5})$	None	$0$	$2$	$-2$	$0$	$q+(1+\beta _{1}+\beta _{2})q^{3}+(\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots$
1568.2.i.w	$4$	$12.521$	$\Q(\sqrt{-3}, \sqrt{5})$	None	$0$	$2$	$2$	$0$	$q+(1+\beta _{1}+\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{5}+\cdots$
1568.2.i.x	$4$	$12.521$	$\Q(\sqrt{2}, \sqrt{-3})$	None	$0$	$2$	$2$	$0$	$q+(1+\beta _{1}+\beta _{2})q^{3}+(-2\beta _{1}-\beta _{2}-2\beta _{3})q^{5}+\cdots$
1568.2.i.y	$8$	$12.521$	8.0.207360000.1	None	$0$	$0$	$0$	$0$	$q-\beta _{2}q^{3}+(-2\beta _{1}+2\beta _{4})q^{5}-7\beta _{3}q^{9}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(1568, [\chi])$ into lower level spaces

Overview	Random
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 25 \)
Sturm bound:			\(448\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(1568, [\chi])\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(1568, [\chi])\) into lower level spaces

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1568.2.i

Level	$1568$
Weight	$2$
Character orbit	1568.i
Rep. character	$\chi_{1568}(961,\cdot)$
Character field	$\Q(\zeta_{3})$
Dimension	$80$
Newform subspaces	$25$
Sturm bound	$448$
Trace bound	$25$

Label	Dim.	\(A\)	Field	CM	Traces				$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	$q$-expansion
1568.2.i.a	\(2\)	\(12.521\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(-2\)	\(-2\)	\(0\)	\(q+(-2+2\zeta_{6})q^{3}-2\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots\)
1568.2.i.b	\(2\)	\(12.521\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(-2\)	\(0\)	\(0\)	\(q+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots\)
1568.2.i.c	\(2\)	\(12.521\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(-2\)	\(0\)	\(0\)	\(q+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots\)
1568.2.i.d	\(2\)	\(12.521\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(-2\)	\(2\)	\(0\)	\(q+(-2+2\zeta_{6})q^{3}+2\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots\)
1568.2.i.e	\(2\)	\(12.521\)	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-1}) \)	\(0\)	\(0\)	\(-4\)	\(0\)	\(q-4\zeta_{6}q^{5}+3\zeta_{6}q^{9}-4q^{13}+(-8+\cdots)q^{17}+\cdots\)
1568.2.i.f	\(2\)	\(12.521\)	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-1}) \)	\(0\)	\(0\)	\(-2\)	\(0\)	\(q-2\zeta_{6}q^{5}+3\zeta_{6}q^{9}-6q^{13}+(2-2\zeta_{6})q^{17}+\cdots\)
1568.2.i.g	\(2\)	\(12.521\)	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-1}) \)	\(0\)	\(0\)	\(2\)	\(0\)	\(q+2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+6q^{13}+(-2+\cdots)q^{17}+\cdots\)
1568.2.i.h	\(2\)	\(12.521\)	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-1}) \)	\(0\)	\(0\)	\(4\)	\(0\)	\(q+4\zeta_{6}q^{5}+3\zeta_{6}q^{9}+4q^{13}+(8-8\zeta_{6})q^{17}+\cdots\)
1568.2.i.i	\(2\)	\(12.521\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(2\)	\(-2\)	\(0\)	\(q+(2-2\zeta_{6})q^{3}-2\zeta_{6}q^{5}-\zeta_{6}q^{9}+(4+\cdots)q^{11}+\cdots\)
1568.2.i.j	\(2\)	\(12.521\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(2\)	\(0\)	\(0\)	\(q+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots\)
1568.2.i.k	\(2\)	\(12.521\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(2\)	\(0\)	\(0\)	\(q+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots\)
1568.2.i.l	\(2\)	\(12.521\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(2\)	\(2\)	\(0\)	\(q+(2-2\zeta_{6})q^{3}+2\zeta_{6}q^{5}-\zeta_{6}q^{9}+(-4+\cdots)q^{11}+\cdots\)
1568.2.i.m	\(4\)	\(12.521\)	\(\Q(\sqrt{-3}, \sqrt{5})\)	None	\(0\)	\(-2\)	\(-2\)	\(0\)	\(q+(-1-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots\)
1568.2.i.n	\(4\)	\(12.521\)	\(\Q(\sqrt{-3}, \sqrt{5})\)	None	\(0\)	\(-2\)	\(2\)	\(0\)	\(q+(-1-\beta _{1}-\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
1568.2.i.o	\(4\)	\(12.521\)	\(\Q(\sqrt{2}, \sqrt{-3})\)	None	\(0\)	\(-2\)	\(2\)	\(0\)	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(2\beta _{1}-\beta _{2}+2\beta _{3})q^{5}+\cdots\)
1568.2.i.p	\(4\)	\(12.521\)	\(\Q(\sqrt{-3}, \sqrt{7})\)	None	\(0\)	\(0\)	\(-6\)	\(0\)	\(q+\beta _{1}q^{3}+3\beta _{2}q^{5}+4\beta _{2}q^{9}+\beta _{1}q^{11}+\cdots\)
1568.2.i.q	\(4\)	\(12.521\)	\(\Q(\sqrt{2}, \sqrt{-3})\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(q+\beta _{1}q^{3}-\beta _{2}q^{9}+(-2-2\beta _{2})q^{11}+\cdots\)
1568.2.i.r	\(4\)	\(12.521\)	\(\Q(\sqrt{2}, \sqrt{-3})\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(q+\beta _{1}q^{3}-\beta _{2}q^{9}+(2+2\beta _{2})q^{11}-2\beta _{3}q^{13}+\cdots\)
1568.2.i.s	\(4\)	\(12.521\)	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-1}) \)	\(0\)	\(0\)	\(0\)	\(0\)	\(q+3\beta _{1}q^{5}+(3+3\beta _{2})q^{9}+5\beta _{3}q^{13}+\cdots\)
1568.2.i.t	\(4\)	\(12.521\)	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-1}) \)	\(0\)	\(0\)	\(0\)	\(0\)	\(q-\beta _{1}q^{5}+(3+3\beta _{2})q^{9}+\beta _{3}q^{13}+(-5\beta _{1}+\cdots)q^{17}+\cdots\)
1568.2.i.u	\(4\)	\(12.521\)	\(\Q(\zeta_{12})\)	None	\(0\)	\(0\)	\(2\)	\(0\)	\(q-\zeta_{12}^{2}q^{3}+(1-\zeta_{12})q^{5}+3\zeta_{12}^{2}q^{11}+\cdots\)
1568.2.i.v	\(4\)	\(12.521\)	\(\Q(\sqrt{-3}, \sqrt{5})\)	None	\(0\)	\(2\)	\(-2\)	\(0\)	\(q+(1+\beta _{1}+\beta _{2})q^{3}+(\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots\)
1568.2.i.w	\(4\)	\(12.521\)	\(\Q(\sqrt{-3}, \sqrt{5})\)	None	\(0\)	\(2\)	\(2\)	\(0\)	\(q+(1+\beta _{1}+\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{5}+\cdots\)
1568.2.i.x	\(4\)	\(12.521\)	\(\Q(\sqrt{2}, \sqrt{-3})\)	None	\(0\)	\(2\)	\(2\)	\(0\)	\(q+(1+\beta _{1}+\beta _{2})q^{3}+(-2\beta _{1}-\beta _{2}-2\beta _{3})q^{5}+\cdots\)
1568.2.i.y	\(8\)	\(12.521\)	8.0.207360000.1	None	\(0\)	\(0\)	\(0\)	\(0\)	\(q-\beta _{2}q^{3}+(-2\beta _{1}+2\beta _{4})q^{5}-7\beta _{3}q^{9}+\cdots\)